Information Qecthreshold From Phi Ladder Code Threshold Pos

The positivity guarantee

Error-correcting codes protect information by spreading it across many physical bits or qubits. Each family of codes has a threshold: a noise rate below which errors can be corrected and above which they cannot. The repetition code, which stores one bit in several copies, tolerates around 50 percent noise. The surface code, a two-dimensional grid of qubits, tolerates about 1 percent. These thresholds are small positive fractions, and the question is whether anything structural forces them to stay that way.

In Recognition Science, the framework defines a threshold as a power of the golden ratio phi, the number satisfying r² = r + 1, approximately 1.618. The definition is simple: for a natural number k, the threshold is phi raised to the power k, then inverted. For k = 9, this gives phi⁻⁹, about 0.013, matching the surface code's roughly 1.3 percent threshold. The framework's machine-checked library of formal theorems proves that this quantity is always positive for every natural number k. The proof is short: phi is positive, a positive number raised to any power is positive, and the reciprocal of a positive number is positive. That is the entire content of the declaration codeThreshold_pos.

The declaration does not claim that any real code family actually achieves these thresholds, nor that the golden ratio is the only possible source of thresholds. It does not assert that the five named code families, repetition, surface, colour, topological, and concatenated, are the only ones that exist. Those five are a definitional choice in the framework, not a proved fact about the physical world. The theorem only guarantees that the framework's own threshold function stays in the positive numbers, which is a necessary sanity condition for any quantity meant to represent a noise rate.

What the positivity guarantee changes is simple: it rules out a whole class of nonsense before it starts. A threshold of zero would mean no noise can ever be corrected, and a negative threshold would be meaningless as a rate. By proving positivity, the framework ensures its error-correction model never produces such degenerate values. The further claim, that adjacent code families differ by a factor of phi, is a separate theorem about the ratio between consecutive thresholds, and it depends on this positivity result to avoid division by zero.

noncomputable def codeThreshold (k : ℕ) : ℝ := (phi ^ k)⁻¹

theorem codeThreshold_pos (k : ℕ) : 0 < codeThreshold k :=
  inv_pos.mpr (pow_pos phi_pos k)

What this page does not claim

No real error-correcting code is proven to achieve the phi-power thresholds. The five named code families are a definitional choice, not a proved exhaustive list. The positivity theorem says nothing about the numerical value of any threshold, only that it is greater than zero.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Information/QECThresholdFromPhiLadder.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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